Krull-Schmidt Fails for Artinian Modules 

by Alberto Facchini, Dolors Herbera, Lawrence S. Levy, Peter V\'amos

KEYWORDS 
        Krull-Schmidt, artinian module, direct sum decomposition, 
endomorphism ring 
DATE 
        10/13/95 
STATUS 
        Proceedings American Math Society 123 (1995) pp. 3587 - 3592, copyright by the AMS,
        1995 
COMMENT 
        Typset in AMS-Latex 2.09 



Abstract

We prove that the Krull-Schmidt theorem fails for artinian modules. This 
answers a question asked by Krull in 1932. In fact we show that if $S$ is a 
module-finite algebra over a semilocal noetherian commutative ring, then 
every nonunique decomposition of every noetherian $S$-module $M$ leads to an 
analogous nonunique decomposition of an artinian module over a related 
non-noetherian ring. The key to this is that the endomorphism ring of $M$ 
is also the endomorphism ring of some artinian module.